As the swim coach at Almond, Christopher selects which athletes will participate in the state-wide swim relay. The relay team swims $\frac{4}{5}$ of a mile in total, with each team member responsible for swimming $\frac{4}{15}$ of a mile. The team must complete the swim in $\frac{3}{4}$ of an hour. How many swimmers does Christopher need on the relay team?
Solution: To find out how many swimmers Christopher needs on the team, divide the total distance ( $\frac{4}{5}$ of a mile) by the distance each team member will swim ( $\frac{4}{15}$ of a mile). $ \dfrac{{\dfrac{4}{5} \text{ mile}}} {{\dfrac{4}{15} \text{ mile per swimmer}}} = {\text{ number of swimmers}} $ Dividing by a fraction is the same as multiplying by the reciprocal. The reciprocal of ${\dfrac{4}{15} \text{ mile per swimmer}}$ is ${\dfrac{15}{4} \text{ swimmers per mile}}$ $ {\dfrac{4}{5}\text{ mile}} \times {\dfrac{15}{4} \text{ swimmers per mile}} = {\text{ number of swimmers}} $ $ \dfrac{{4} \cdot {15}} {{5} \cdot {4}} = {\text{ number of swimmers}} $ Reduce terms with common factors by dividing the $4$ in the numerator and the $4$ in the denominator by $4$ $ \dfrac{{\cancel{4}^{1}} \cdot {15}} {{5} \cdot {\cancel{4}^{1}}} = {\text{ number of swimmers}} $ Reduce terms with common factors by dividing the $15$ in the numerator and the $5$ in the denominator by $5$ $ \dfrac{{1} \cdot {\cancel{15}^{3}}} {{\cancel{5}^{1}} \cdot {1}} = {\text{ number of swimmers}} $ Simplify: $ \dfrac{{1} \cdot {3}} {{1} \cdot {1}} = {3} $ Christopher needs 3 swimmers on his team.